Performance Analysis of Effective Methods for Solving Band Matrix SLAEs After Parabolic Nonlinear PDEs

Abstract

This paper presents an experimental performance study of implementations of three different types of algorithms for solving band matrix systems of linear algebraic equations (SLAEs) after parabolic nonlinear partial differential equations – direct, symbolic, and iterative, the former two of which were introduced in Veneva and Ayriyan in Effective methods for solving band SLAEs after parabolic nonlinear PDEs (2018) [3]. An iterative algorithm is presented – the strongly implicit procedure (SIP), also known as the Stone method. This method uses the incomplete LU (ILU(0)) decomposition. An application of the Hotelling-Bodewig iterative algorithm is suggested as a replacement of the standard forward-backward substitutions. The upsides and the downsides of the SIP method are discussed. The complexity of all the investigated methods is presented. Performance analysis of the implementations is done using the high-performance computing (HPC) clusters “HybriLIT” and “Avitohol”. To that purpose, the experimental setup and the results from the conducted computations on the individual computer systems are presented and discussed.

Appendix

The Hotelling-Bodewig iterative algorithm has the form as follows:

$$\begin{aligned} A^{-1}_{n+1} = A^{-1}_{n}\,(2\,I - A\,A^{-1}_{n}), \quad n=0,1,\ldots , \end{aligned}$$

(1)

where I is the identity matrix, A is the matrix whose inverse we are looking for. $A^{-1}_{0}$ is taken to be of a diagonal form.

The obtained computational times for the ILU(0) method, the Hotelling-Bodewig iterative algorithm and the Stone method, using the heterogeneous cluster “HybriLIT” and the supercomputer system “Avitohol”, are summarized in Tables 8, 9, and 10.

Table 8 Results from the ILU(0) method and the numerical method for inverting matrices, using the cluster “HybriLIT”

Table 9 Results from the ILU(0) method and the numerical method for inverting matrices, using the cluster “Avitohol”

Table 10 Results from solving a SLAE using SIP on the clusters “HybriLIT” and “Avitohol”

The matrix implementations lead to 5, 7, and 34 iterations, respectively for finding $L^{-1}$ and $U^{-1}$, applying the Hotelling-Bodewig procedure, and for the Stone method while the needed iterations when the array implementations are executed are 5, 6, and 31, respectively. It is expected that inverting L would require less number of iterations, since it is a unit triangular matrix. The achieved accuracy is of an order of magnitude of $10^{-13}$, having used an error tolerance $10^{-12}$. Comparing the results for the computational times, one can see that the array implementation not only decreased the time needed for the inversion of both the matrices L and U but also it decreases the number of iterations needed so as the matrix U to be inverted. As one can see, the time required for the SIP procedure is also improved by the new implementation approach. One reason being is that the number of iterations is decreased. Overall, the array implementations decrease the computational times with one order of magnitude. Finally, this second approach requires less amount of memory (instead of keeping $N\times N$ matrix, just 5 arrays with length N are stored), which allows experiments with bigger matrices to be conducted. However, this method (even in its array form) is not suitable for too large matrices (with number of rows bigger than $1\times 10^5$), since the evaluation of the inverse of a matrix is computationally demanding on both time and memory. A comparison between the times on the two computer systems showed that overall “HybriLIT” is a bit faster than “Avitohol”.